Welcome to Math Mutation, the podcast where we discuss fun, interesting, or weird corners of mathematics that you would not have heard in school. Recording from our headquarters in the suburbs of Wichita, Kansas, this is Erik Seligman, your host. And now, on to the math.
Mazes, or labyrinths, are probably the one type of somewhat mathematical game that nearly everyone is familiar with. Books of activities for young children, at least in the US, often contain simple mazes, drawings with many different twisty passages leading from one side of the page to the other, with a challenge to trace the right path. And when I was first learning to program computers while growing up, one of my favorite types of simple programs to write and experiment with were ones that generate and draw mazes and labyrinths. As you are likely aware, these have been around in both literary and visual form since ancient times.
Recently I was reading a book on this topic by a scholar named Penelope Doob, and she pointed out an intriguing mystery related to these constructs. There are two major types of labyrinths: unicursal ones, where there is ultimately only one path (though a very twisty one) that will eventually lead anyone entering to the goal; and multicursal ones, where you have to make choices along the way among many paths, most of which lead to dead ends. The word “labyrinth” can refer to both, though these days “maze” usually refers to the multicursal type. But here’s the strange thing: in ancient and medieval sources, nearly every visual or physically built maze is unicursal or single-path, while nearly every literary description of labyrinths describes a multicursal or multiple-path one. Why would we have this strange division? Why weren’t people drawing and building the same labyrinths they described in their stories?
Let’s start by looking in more detail at unicursal labyrinths, the ones with only one path from beginning to end. This one path is guaranteed to get anyone entering to their goal eventually, but it will take a lot longer than one might guess from the size of the labyrinth. This is often seen as a spiritual metaphor: the path to true faith or Enlightenment may be confusing and long, but as long as you remain steady and stick to the path, following the prescribed moral choices that move you forward, you will reach your goal. Similarly, when laid into the ground or in a garden, it can be seen as a meditation aid, providing a simple path to slowly follow which doesn’t require any choices or decisions, and helping to free your mind of conscious thought. Actually, that’s not fully true: you do have the initial choice to enter the labyrinth or not, as well as the choice to turn back at any time. But this also can work well with the metaphor of following the steady path to God: you make the critical choice of whether to embark on the path, and at any moment along the way, you either maintain your faith or turn away. As mentioned above, nearly all visually depicted labyrinths in ancient and medieval times are unicursal, with only a handful of exceptions. Doob’s book shows some interesting examples dating back to stone carvings between 1800-1400 BC.
The multicursal type of labyrinth, where you have many choices to make along the way and can end up in a dead end or trap, is the kind most suited to games and puzzles. If you’re a modern gamer, I’m sure you've wandered around an endless set of 3-D mazes in various videogames or in tabletop games like Dungeons & Dragons. The literary precedents for this type of labyrinth go back to ancient times as well— who can forget the Greek legend of the Minotaur, who lived at the center of a confusing and dangerous labyrinth. Theseus only managed to make it out alive because the princess Ariadne gave him some thread which he could slowly unwind to mark his path. These labyrinths can also provide a slightly different metaphor for the path to religious enlightenment: the constant presence of temptations to sin, which will lead you away from your goal, possibly forever. Whether you succeed and make it to the goal or spend the rest of your life wandering in confusion, it’s due to your own choices.
But we should not forget that there are a number of elements that both types of labyrinths have in common. When you look at either a unicursal or multicursal maze from above, they are a visually busy picture that’s hard to fully comprehend at a glance; in most cases, you can’t even be sure a labyrinth is of one type or the other without starting to carefully trace the path. Both seem to trigger the same set of concepts in our brain, at least at first glance. Also in both cases, because the maze twists back and forth many times within a small confined area, it’s very hard to know how far along the path you are: while the unicursal maze might not provide false paths, it still conceals the total distance to the goal. So they both have the same effects in attacking your self-confidence and triggering confusion, These commonalities are strong enough that the ancients generally used the same language to describe both, variants of the words that led to our modern “labyrinth”.
Ultimately, these commonalities provide a key to understanding the strange issue of visually depicted unicursal mazes and literary multicursal ones. As Doob describes it, “the best solution that can be found to the mystery is that classical and medieval eyes saw insufficient difference in the implications of the two models to warrant a new design.” Unicursal mazes are somewhat simpler to draw or illustrate, since you don’t have to worry about accidentally failing to create a non-dead-end path to the goal And it was very common for most art to be based somewhat on earlier art, so once drawings of unicursal labyrinths became common, those who needed to draw something similar followed the patterns set by their predecessors. When artists were creating illustrations of labyrinths, they didn’t consult the corresponding literary works to check for an exact match. On the other hand, one cannot deny that when telling stories, multicursal mazes are much more terrifying, providing the possibility of getting lost or confused forever without reaching your goal, so it makes a lot of sense that these would feature prominently in myths and legends.
And this has been your math mutation for today.
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